2.2 (Local linearization around a trajectory). A single-wheel cart (unicycle) mov- ing on the plane with linear velocity v and angular velocity @ can be modeled by the nonlinear system Px = v cose, Py = v sin 0, ė = w, (2.11) where (px, py) denote the Cartesian coordinates of the wheel and its orientation. Regard this as a system with input u := [v_ @]' = R². (a) Construct a state-space model for this system with state x = X1 X2 := X3 Px cose + (py - 1) sin -Px sin0 + (py - 1) cos Ө and output y := [x₁ x₂] € R². = (b) Compute a local linearization for this system around the equilibrium point xºq = 0, ueq = 0. (c) Show that w (t) = v(t) = 1, px(t) = sint, py(t) = 1 − cost, 0(t) = t, \t ≥ 0 is a solution to the system. (d) Show that a local linearization of the system around this trajectory results in an LTI system.

2.2 (Local linearization around a trajectory). A single-wheel cart (unicycle) mov- ing on the plane with linear velocity v and angular velocity @ can be modeled by the nonlinear system Px = v cose, Py = v sin 0, ė = w, (2.11) where (px, py) denote the Cartesian coordinates of the wheel and its orientation. Regard this as a system with input u := [v_ @]' = R². (a) Construct a state-space model for this system with state x = X1 X2 := X3 Px cose + (py - 1) sin -Px sin0 + (py - 1) cos Ө and output y := [x₁ x₂] € R². = (b) Compute a local linearization for this system around the equilibrium point xºq = 0, ueq = 0. (c) Show that w (t) = v(t) = 1, px(t) = sint, py(t) = 1 − cost, 0(t) = t, \t ≥ 0 is a solution to the system. (d) Show that a local linearization of the system around this trajectory results in an LTI system.

Power System Analysis and Design (MindTap Course List)

6th Edition

ISBN:9781305632134

Author:J. Duncan Glover, Thomas Overbye, Mulukutla S. Sarma

Publisher:J. Duncan Glover, Thomas Overbye, Mulukutla S. Sarma

Chapter6: Power Flows

Section: Chapter Questions

Problem 6.27P

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